The tail of a quantum spin network

TL;DR

This paper investigates the tail of sequences of colored trivalent graphs using skein relations, deriving product formulas and connecting these to classical identities like the Andrews-Gordon identities and Ramanujan theta functions.

Contribution

It introduces skein-theoretic methods to compute tails of colored trivalent graphs and links these computations to well-known q-series identities.

Findings

Derived product formulas for tails of colored trivalent graphs

Connected skein techniques to Andrews-Gordon and Ramanujan identities

Provided a new proof for classical q-series identities

Abstract

The tail of a sequence $\{P_n(q)\}_{n \in \mathbb{N}}$ of formal power series in $\mathbb{Z}[[q]]$ is the formal power series whose first $n$ coefficients agree up to a common sign with the first $n$ coefficients of $P_n$. This paper studies the tail of a sequence of admissible trivalent graphs with edges colored $n$ or $2n$. We use local skein relations to understand and compute the tail of these graphs. We also give product formulas for the tail of such trivalent graphs. Furthermore, we show that our skein theoretic techniques naturally lead to a proof for the Andrews-Gordon identities for the two variable Ramanujan theta function as well to corresponding identities for the false theta function.